The Ideal Gas Law

PV = nRT relates the four state variables of a gas sample. Use R = 0.08206 Latm/(molK) when pressure is in atmospheres, or R = 8.314 J/(mol*K) when you need energy units. Temperature is always in kelvin, obtained by adding 273.15 to a Celsius value.

Because 1 L kPa = 1 J, the numerical value 8.314 works for both SI pressure-volume products listed. It does not work with atm and L without conversion. Use absolute pressure rather than a gauge pressure relative to the atmosphere. If a gauge reads 1.00 atm above an ambient 1.00 atm, the absolute pressure is 2.00 atm.

Hold n and T constant and PV is a constant, which is the law of Boyle. Hold n and P constant and V/T is a constant, which is the law of Charles. Hold P and T constant and V/n is a constant, which is the law of Avogadro. You never need to memorise them separately.

Write PV/(nT) = R before deciding what cancels. With n and T fixed, P₁V₁ = P₂V₂. With n and P fixed, V₁/T₁ = V₂/T₂. With P and T fixed, V₁/n₁ = V₂/n₂. If the problem changes temperature as well as pressure, the one-variable version is no longer sufficient. Gas volume is proportional to kelvin temperature, not Celsius temperature, because the absolute scale tracks thermal energy.

When the same sample moves between two states, divide one form of the equation by the other. Anything held constant cancels, leaving P₁V₁/T₁ = P₂V₂/T₂. If moles also change, keep n in the expression as P₁V₁/(n₁T₁) = P₂V₂/(n₂T₂).

For a fixed sample initially at 1.00 atm, 2.00 L and 300 K, find its volume at 0.800 atm and 360 K. Rearranging gives V₂ = V₁(P₁/P₂)(T₂/T₁) = 2.00(1.00/0.800)(360/300) = 3.00 L. Both lower pressure and higher temperature favor expansion, so the direction checks. If the vessel is rigid, volume is constrained instead; do not solve for an expansion that the apparatus prevents. If gas escapes, n is not constant and must be retained.

Substituting n = m/M into PV = nRT gives PM = dRT, where d is density in grams per litre. This is how a gas density measurement identifies an unknown gas, and it explains why warm air, at lower density, rises.

Density d is m/V. Substituting n = m/M into PV = nRT and dividing by volume gives P = dRT/M, so d = PM/(RT). At equal pressure and temperature, a gas with larger molar mass has greater density. A measured molar mass of about 44 g/mol is consistent with CO₂ but does not uniquely identify it; other gases or mixtures can have similar mean molar masses.

Before solving, list P, V, n and T, mark the unknown, convert T to kelvin and choose a consistent R. Estimate the direction of change, then evaluate and check the output units. The ideal approximation is most useful at sufficiently low density and away from condensation.

“STP” has more than one pressure convention. At 273.15 K and 1 atm, the ideal molar volume is about 22.414 L/mol; at 273.15 K and 1 bar it is about 22.711 L/mol. Specify the convention instead of treating 22.4 L/mol as a universal constant. This course's 22.4 shortcut uses 0 °C and 1 atm. At room temperature, calculate RT/P rather than using that number.